Abstract:
This thesis gives a detailed account of James’ famous characterisation of weak compactness, originally from [15]. We give a full proof of James’ Theorem, and then prove a number of related problems using the same techniques. Firstly, we generalise James’ result to a statement about the subdifferential of a continuous, convex function. Then, we show how James’ original approach can be used to prove a result about the notion of (I)-generation from [23]. Lastly, we present results about how the topology of the set of all norm-attaining functionals characterises reflexivity. Our approach combines the work of James with the work of Jimenez Sevilla and Moreno from [17].